Finite Group Databasebeta

← #30638693

Subgroups — #30638693

Order 16256.

548 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550712C211yesmaximal, characteristic
2#22549892C212yesmaximal, complemented
3#22549873C212yesmaximal, complemented
4#22549850C211yesmaximal, characteristic, complemented
5#408×C127C211yesmaximal, characteristic, complemented
6#16861257C412yescomplemented
7#16861256—22no—
8#16861255—22no—
9#16861254—22no—
10#16861254C412yescomplemented
11C2×#14523248—22no—
12C2×#14523248C2212yes—
13#41×D127—24no—
14#16861168C2212yescomplemented
15#16861135—24no—
16#16861134—22no—
17#16861131—22no—
18#16861124—22no—
19C2×#41×C127C2211yescharacteristic
20C2×M16×C127C2211yescharacteristic, complemented
21C22×C8×C127C2211yescharacteristic, complemented
22C22⋊C8×C127—22no—
23#14523249—24no—
24#14523248—24no—
25#14523248—24no—
26#14523248—24no—
27#14523248—22no—
28#14523248—22no—
29#14523248C2×C414yes—
30#14523248D414yes—
31#14523247—28no—
32#14523247—24no—
33C2×Dic254—22no—
34C2×Dic254D412yes—
35Q8×D127—24no—
36C2×D508—22no—
37C2×D508D412yes—
38C2×C22⋊D127—24no—
39C2×C22⋊D127—22no—
40D4×D127—28no—
41D4×D127—24no—
42C2×C4×D127—24no—
43C2×C4×D127—22no—
44C4⋊Dic127—22no—
45C22⋊Dic127—22no—
46C4×Dic127—22no—
47#41×C127—22no—
48#41×C127—22no—
49C2×Q8×C127C2×C411yescharacteristic
50C2×D4×C127C2×C411yescharacteristic
51C2×D4×C127D412yes—
52C22×C4×C127D412yes—
53C22×C4×C127C2311yescharacteristic
54M16×C127—22no—
55C2×C8×C127—21no—
56C2×C8×C127—21no—
57C2×C8×C127—22no—
58C22×D127—24no—
59C22×D127—22no—
60Dic254—24no—
61Dic254—22no—
62Dic254C22⋊C414yes—
63D508—24no—
64D508—22no—
65D508C22⋊C414yes—
66C22⋊D127—28no—
67C22⋊D127—24no—
68C22⋊D127—24no—
69C2×Dic127—24no—
70C2×Dic127—22no—
71C2×Dic127—42no—
72C4×D127—28no—
73C4×D127—24no—
74C4×D127—24no—
75C23×C127C22⋊C412yes—
76C23×C127C2×D411yescharacteristic
77Q8×C127—22no—
78D4×C127—24no—
79D4×C127—22no—
80C2×C4×C127—22no—
81C2×C4×C127—22no—
82C2×C4×C127C22⋊C412yes—
83C2×C4×C127C22×C411yescharacteristic
84C2×C4×C127C2×D412yes—
85C2×C4×C127C2×D412yes—
86C2×C4×C127C2×D411yescharacteristic
87C8×C127—41no—
88C8×C127—41no—
89C22×C127—22no—
90C22×C127—41no—
91C22×C127—42no—
92C22×C127C2×C22⋊C411yescharacteristic
93C22×C127#12012yes—
94C2×D127—24no—
95C2×D127—82no—
96C2×D127—84no—
97C4×C127—22no—
98C4×C127C2×C22⋊C412yes—
99C4×C127#12011yescharacteristic
100C4×C127#12011yescharacteristic
101Dic127—24no—
102Dic127—82no—
103C2×C127—22no—
104C2×C127—21no—
105C2×C127—42no—
106C2×C127#37911yescharacteristic
107D127—44no—
108#1794—1271nomaximal, Sylow
109C127#179411yescharacteristic, complemented, Sylow
110#585—1271no—
111#421—1272no—
112#416—1272no—
113#410—1271no—
114#408D12711yescharacteristic, complemented
1152−1+4—1272no—
1162−1+4—2542no—
117H(5,𝔽2)—1272no—
118H(5,𝔽2)—2542no—
119H(5,𝔽2)—2542no—
120C2×#41—1272no—
121C2×#41—2544no—
122C2×#41—2542no—
123C2×#41C2×D12711yescharacteristic
124C2×M16C2×D12711yescharacteristic, complemented
125C22×C8C2×D12711yescharacteristic, complemented
126#117—1272no—
127#104—2544no—
128#103—2542no—
129#102—2542no—
130#99—2542no—
131C22⋊C8—22no—
132#41—22no—
133#41—22no—
134#41—1274no—
135#41—1274no—
136#41—2544no—
137#41—2544no—
138#41—2544no—
139#41—2548no—
140#41—2544no—
141#41—2544no—
142#41—2542no—
143#41—2542no—
144C2×Q8—1272no—
145C2×Q8—2542no—
146C2×Q8—2544no—
147C2×Q8C4×D12711yescharacteristic
148C2×D4—1272no—
149C2×D4—2542no—
150C2×D4—2544no—
151C2×D4—2548no—
152C2×D4—2544no—
153C2×D4—2542no—
154C2×D4C4×D12711yescharacteristic
155C2×D4D50812yes—
156C22×C4—2544no—
157C22×C4—2542no—
158C22×C4C22⋊D12712yes—
159C22×C4C22×D12711yescharacteristic
160M16—22no—
161C2×C8—22no—
162C2×C8—21no—
163C2×C8—21no—
164C4⋊C4—2542no—
165C22⋊C4—2542no—
166C42—2542no—
167C23—2544no—
168C23—2542no—
169C23(C2×D127)⋊C412yes—
170C23C2×D50811yescharacteristic
171Q8—22no—
172Q8—1274no—
173Q8—2544no—
174Q8—2542no—
175D4—24no—
176D4—22no—
177D4—1274no—
178D4—2544no—
179D4—2548no—
180D4—2544no—
181D4—2544no—
182D4—2542no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—5082no—
186C2×C4—2548no—
187C2×C4—2544no—
188C2×C4—2542no—
189C2×C4—2544no—
190C2×C4—2544no—
191C2×C4(C2×D127)⋊C412yes—
192C2×C4C2×C4×D12711yescharacteristic
193C2×C4D4×D12712yes—
194C2×C4D4×D12712yes—
195C2×C4C2×C22⋊D12711yescharacteristic
196C8—41no—
197C8—41no—
198C22—22no—
199C22—41no—
200C22—42no—
201C22—10162no—
202C22—10164no—
203C22—2544no—
204C22C2×(C2×D127)⋊C411yescharacteristic
205C22#1686117912yes—
206C4—22no—
207C4—10162no—
208C4—2544no—
209C4C22⋊C4×D12712yes—
210C4#1686117711yescharacteristic
211C4#1686117711yescharacteristic
212C2—22no—
213C2—21no—
214C2—42no—
215C2—5084no—
216C2#2254973111yescharacteristic
subgroupG/H
centerC4#16861177
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#408×C127C2
socleC2×C127#379
prime radical#22550712C2
cosocleC22×C4×C127C23
hypercenter#408D127
hypocenterC127#1794
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #408

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #22549892C2semidirect
3. #22549873C2semidirect
4. #22549850C2semidirect
5. #408×C127C2semidirect
6. #16861257C4semidirect
10. #16861254C4semidirect
14. #16861168C22semidirect
20. C2×M16×C127C22semidirect
21. C22×C8×C127C22semidirect
109. C127#1794semidirect
114. #408D127semidirect
124. C2×M16C2×D127semidirect
125. C22×C8C2×D127semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225507122
#225498922
#225498732
#225498502
#408×C1272
#1794127